least cost path problem in the presence of congestion* #
DESCRIPTION
Least Cost Path Problem in the Presence of Congestion* #. Avijit Sarkar Assistant Professor School of Business University of Redlands. * This is joint work with Drs. Rajan Batta & Rakesh Nagi, Department of Industrial Engineering, SUNY at Buffalo - PowerPoint PPT PresentationTRANSCRIPT
MS-GIS colloquium: 9/28/05
Least Cost Path Problem in the Presence of Congestion*#
Avijit SarkarAssistant ProfessorSchool of Business
University of Redlands
* This is joint work with Drs. Rajan Batta & Rakesh Nagi, Department of Industrial Engineering, SUNY at Buffalo
# Submitted to European Journal of Operations Research
2 of 32MS-GIS colloquium: 9/28/05
2005 Urban Mobility Study http://mobility.tamu.edu/
3 of 32MS-GIS colloquium: 9/28/05
Traffic Mobility Data for 2003 http://mobility.tamu.edu/
4 of 32MS-GIS colloquium: 9/28/05
Traffic Mobility Data for Riverside-San Bernardino, CA http://mobility.tamu.edu/
5 of 32MS-GIS colloquium: 9/28/05
How far has congestion spread?http://mobility.tamu.edu/
Some Results 2003 1982
# of urban areas with TTI > 1.30 28 1
Percentage of traffic experiencing peak period travel congestion
67 32
Percentage of major road system congestion 59 34
# of hours each day when congestion is encountered
7.1 4.5
6 of 32MS-GIS colloquium: 9/28/05
Travel Time Index Trends http://mobility.tamu.edu/
7 of 32MS-GIS colloquium: 9/28/05
Congested Regions – Definition and Details
Urban zones where travel times are greatly increasedClosed and bounded area in the planeApproximated by convex polygonsPenalizes travel through the interior Congestion factor α Cost inside = (1+α)x(Cost Outside) 0 < α < ∞
Shortest path ≠ Least Cost Path Entry/exit point Point at which least cost path enters/exits a congested region Not known a priori
8 of 32MS-GIS colloquium: 9/28/05
Least Cost PathsEfficient route => determine rectilinear least cost paths in the presence of
congested regions
9 of 32MS-GIS colloquium: 9/28/05
Previous Results (Butt and Cavalier, Socio-Economic Planning Sciences, 1997)
Planar p-median problem in the presence of congested regions
Least cost coincides with easily identifiable grid
Imprecise result: holds for rectangular congested regions
For α=0.30, cost=14
For α=0.30, cost=13.8
10 of 32MS-GIS colloquium: 9/28/05
Mixed Integer Linear Programming (MILP) Approach to Determine Entry/Exit Points
(4,3)
P (9,10)
11 of 32MS-GIS colloquium: 9/28/05
MILP Formulation (Sarkar, Batta, Nagi: Socio Economic Planning Sciences: 38(4), Dec 04)
otherwise
CRofiedgeonliesEEEifuzw
Mzyyxxyyxx
xxx
cybxa
iuzw
iMwxx
iMwxx
iMuxx
iMuxx
iMzxx
iMzxx
www
iMwcybxa
iMwcybxa
uuu
iMucybxa
iMucybxa
zzz
iMzcybxa
iMzcybxa
tosubject
yyxxyyxxyyxxyyxxyyxximize
iii
cccncn
ba
iii
ir
il
ir
il
ir
il
iiii
iiii
iiii
iiii
iiii
iiii
ppnn
,0
,,,1,,
)1(||||||||
0
4,2,11
4,2,1)1(
4,2,1)1(
4,2,1)1(
4,2,1)1(
4,2,1)1(
4,2,1)1(
1
4,2,10)1(
4,2,10)1(
1
4,2,10)1(
4,2,10)1(
1
4,2,10)1(
4,2,10)1(
|||||)||)(|1(|||||)||)(|1(||||min
321
422
4
24242
13
13
12
12
11
11
421
33
33
421
22
22
421
11
11
4443433232121211
Entry point E1 lies on exactly one edge
Exit point E2 lies on exactly one edge
Entry point E3 lies on exactly one edge
Provide bounds on x-coordinates of E1, E2, E3
Final exit point E4 lies on edge 4Takes care of additional distance
12 of 32MS-GIS colloquium: 9/28/05
Results
33.10
33.1 (z = 20)
Entry=(5,4)
Exit=(5,10)
Example: For α=0.30, cost = 2+6(1+0.30)+4 = 13.80
13 of 32MS-GIS colloquium: 9/28/05
Discussion
Formulation outputs Entry/exit points Length of least cost path
Advantages Models multiple entry/exit points Automatic choice of number of entry/exit points Automatic edge selection Break point of α
Disadvantages Generic problem formulation very difficult: due to combinatorics Complexity increases with
Number of sides Number of congested regions
14 of 32MS-GIS colloquium: 9/28/05
Alternative ApproachMemory-based Probing Algorithm
Turning step
15 of 32MS-GIS colloquium: 9/28/05
Why Convexity Restriction?
Approach Determine an upper bound on the number of entry/exit points Associate memory with probes => eliminate turning steps
16 of 32MS-GIS colloquium: 9/28/05
Observation 1: Exponential Number of Staircase Paths may ExistStaircase path:Length of staircase path through p CRs
No a priori elimination possible22p+1 (O(4p)) staircase paths between O and D
|||| DoDoOD yyxxl
p
iODii l
1
O(4p)
17 of 32MS-GIS colloquium: 9/28/05
Exponential Number of Staircase Paths
18 of 32MS-GIS colloquium: 9/28/05
At most Two Entry-Exit Points
61.0
61.0,60.0
,59.00
XE1E2E3E4P
XCBP (bypass)
XCE3E4P
19 of 32MS-GIS colloquium: 9/28/05
3-entry 3-exit does not exist
Compare 3-entry/exit path with 2-entry/exit and 1-entry/exit paths
Proof based on contradiction
Use convexity and polygonal properties
20 of 32MS-GIS colloquium: 9/28/05
Memory-based Probing Algorithm
O
D
21 of 32MS-GIS colloquium: 9/28/05
Memory-based Probing Algorithm
Each probe has associated memory what were the directions of two previous probes?
Eliminates turning stepsUses previous result: upper bound of entry/exit pointsNecessary to probe from O to D and backGenerate network of entry/exit pointsTwo types of arcs: (i) inside CRs (ii) outside CRsSolve shortest path problem on generated network
22 of 32MS-GIS colloquium: 9/28/05
Numerical Results (Sarkar, Batta, Nagi: Submitted to European Journal of Operational Research)
condsseCPU
generatednodesofnumber
ctedterseinCRsofnumber
CRsofnumberp
• Algorithm coded in C
23 of 32MS-GIS colloquium: 9/28/05
Number of CRs Intersected vs Number of Nodes Generated
24 of 32MS-GIS colloquium: 9/28/05
Number of CRs Intersectedvs CPU seconds
25 of 32MS-GIS colloquium: 9/28/05
Summary of Results
O(1.414p) entry/exit points rather than O(4p) in worst case
Works well up to 12-15 CRs
Heuristic approaches for larger problem instances
26 of 32MS-GIS colloquium: 9/28/05
Now the Paradox
Optimal path for α=0.30
27 of 32MS-GIS colloquium: 9/28/05
Known Entry-Exit Heuristic
Entry-exit points are known a priori
Least cost path coincides with an easily identifiable finite grid Convex polygonal restriction no longer necessary
28 of 32MS-GIS colloquium: 9/28/05
Potential Benefits
Refine distance calculation in routing algorithms
Large scale disaster Land parcels (polygons) may be destroyed De-congested routes may become congested Can help
Identify entry/exit points Determine least cost path for rescue teams
Form the basis to solve facility location problems in the presence of congestion
29 of 32MS-GIS colloquium: 9/28/05
Some Issues
Congestion factor has been assumed to be constantIn urban transportation settings α will be time-dependent
Time-dependent shortest path algorithms α will be stochastic
Convex polygonal restrictionCannot determine threshold values of α
30 of 32MS-GIS colloquium: 9/28/05
OR-GIS Models for US Military
UAV routing problem UAVs employed by US military worldwide Missions are extremely dynamic UAV flight plans consider
Time windows Threat level of hostile forces Time required to image a site Bad weather
Surface-to-air threats exist enroute and may increase at certain sites
31 of 32MS-GIS colloquium: 9/28/05
Some Insight into the UAV Routing Problem
Threat zones and threat levels are surrogates for congested regions and congestion factorsDifference: Euclidean distancesObjective: minimize probability of detection in the presence of multiple threat zonesCan assume the probability of escape to be a Poisson random variableBasic result
One threat zone: reduces to solving a shortest path problem Result extends or not for multiple threat zones? Potential application to combine GIS network analysis tools with OR
algorithms
32 of 32MS-GIS colloquium: 9/28/05
Questions